Ken Ward's Mathematics Pages

Vieta's Root Formulas

Vieta's formulas relate the coefficients of a polynomial to its roots.
François Viète (or Vieta), seigneur de la Bigotière, generally known as Franciscus Vieta (1540-1603), was a French mathematician. Naturally, there is some confusion over which name to use!

This page is concerned with the use of Vieta's formulas to examine polynomials and does not deal with the theory.
The formulas are sometimes called Viète's formulas or relations. If the coefficient of xn isn't 1, then we can divide by the coefficient of xn (a_n)
An equation:
xⁿ+a_n-1x^n-1+a_n-2x^n-2... +a₀=0 [1]
can be written:
(x-r₁)(x-r₂)...(x-r_n)=0           [2]
Where r₁, r₂, etc  are the n roots of the equation [1] (This is based on the Fundamental Theorem of Algebra)
By observation, we can see that the sum of the roots is equal to -a_n-1
The sum of the roots is:

And the product is:

Quadratic Formulas

A quadratic equation with roots α and β can be written:
ax²+bx+c=a(x-α)(x-β)=0
For which, Vieta's root forumula is:

Cubic Formulas

For the cubic equation:
ax³+bx²+cx+d=a(x-α)(x-β)(x-γ)=0

Quartic Formulas

For the quartic equation:
ax⁴+bx³+cx²+dx+e-a(x-α)(x-β)(x-γ)(x-δ)=0

Example 1

x²+x-6
If the roots are α and β then, according to Vieta:

α + β=-1 (sum of roots equal to minus coefficient of n-1)
αβ=-6 (product is equal to the  constant term, because n is even)
The roots are -3 and 2, so:
-3+2=-1
(-3)2=-6

Example 2

x³+2x²-5x-6
If the roots are α, β and γ, then according to Vieta:

α+β+γ=-2  (sum of roots equal to minus coefficient of n-1, that is (-1)*2) 
αβγ=6 (product is equal to minus the  constant term (-1)*(-6), because n is odd)

The roots are -3, 2 and -1, so:
-3+2-1=2 
(-3)2(-1)=6

Example 3

x⁴-2x³-13x²+14x+24
If the roots are α, β, γ and δ, then according to Vieta:

α+β+γ+δ=2   (sum of roots equal to minus coefficient of n-1, that is (-1)*(-2))
αβγδ=24      (product is equal to minus the  constant term, because n is even)
The roots are -3, 2 -1 and 4 so:
-3+2-1+4=2
(-3)2(-1)(4)=24

Ken Ward's Mathematics Pages

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Faster Arithmetic - by Ken Ward